Note on the Jordan form of an irreducible eventually nonnegative matrix

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Tam, Bit-Shun
Wilson, Ulrica
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Hogben, Leslie
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Electrical and Computer Engineering

The Department of Electrical and Computer Engineering (ECpE) contains two focuses. The focus on Electrical Engineering teaches students in the fields of control systems, electromagnetics and non-destructive evaluation, microelectronics, electric power & energy systems, and the like. The Computer Engineering focus teaches in the fields of software systems, embedded systems, networking, information security, computer architecture, etc.

The Department of Electrical Engineering was formed in 1909 from the division of the Department of Physics and Electrical Engineering. In 1985 its name changed to Department of Electrical Engineering and Computer Engineering. In 1995 it became the Department of Electrical and Computer Engineering.

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  • Department of Electrical Engineering (1909-1985)
  • Department of Electrical Engineering and Computer Engineering (1985-1995)

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A square complex matrix A is eventually nonnegative if there exists a positive integer k(0) such that for all k >= k(0), A(k) >= 0; A is strongly eventually nonnegative if it is eventually nonnegative and has an irreducible nonnegative power. It is proved that a collection of elementary Jordan blocks is a Frobenius Jordan multiset with cyclic index r if and only if it is the multiset of elementary Jordan blocks of a strongly eventually nonnegative matrix with cyclic index r. A positive answer to an open question and a counterexample to a conjecture raised by Zaslavsky and Tam are given. It is also shown that for a square complex matrix A with index at most one, A is irreducible and eventually nonnegative if and only if A is strongly eventually nonnegative.


This article is published as Hogben, Leslie, Bit-Shun Tam, and Ulrica Wilson. "Note on the Jordan form of an irreducible eventually nonnegative matrix." The Electronic Journal of Linear Algebra 30 (2015): 279-285. DOI: 10.13001/1081-3810.3049. Posted with permission.

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Thu Jan 01 00:00:00 UTC 2015